Optimal bounded control of stochastically excited MDOF nonlinear viscoelastic systems
Optimal bounded control of stochastically excited MDOF nonlinear viscoelastic systems
复制标题
随机激励多自由度非线性粘弹性系统的最优有界控制
DOI：
10.1007/s11071-017-3440-2
复制
发表时间：
2017
影响因子：
5.6
通讯作者：
Zhu W. Q.
中科院分区：
文献类型：
--
作者：
Gu X. D.;Zhu W. Q.
提出了一种基于随机平均法和随机极大值原理的随机激励多自由度非线性粘弹性系统最优有界控制设计新方法。首先，基于准可积哈密顿系统运动的随机周期特性，将系统表述为具有粘弹性项的准可积哈密顿系统，并将每个粘弹性项近似地替换为弹性恢复力和粘阻尼力。从而将随机激励的多自由度非线性粘弹性系统转化为不含粘弹性项的等效拟可积哈密顿系统。然后，通过应用随机平均，将系统进一步简化为低维数的部分平均系统。利用随机极大值原理，导出了部分平均系统最优控制问题的伴随方程和极大值条件，并根据极大值条件确定了最优有界控制力。最后，通过求解与被控系统的完全平均Itô方程相关联的Fokker-Plank-Kolmogorov方程（FPK），得到了最优控制系统平稳响应的概率和统计量。算例说明了该方法的有效性。
A new procedure for designing optimal bounded control of stochastically excited multi-degree-of-freedom (MDOF) nonlinear viscoelastic systems is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the system is formulated as a quasi-integrable Hamiltonian system with viscoelastic terms and each viscoelastic term is replaced approximately by an elastically restoring force and a visco-damping force based on the randomly periodic behavior of the motion of quasi-integrable Hamiltonian system. Thus, a stochastically excited MDOF nonlinear viscoelastic system is converted to an equivalent quasi-integrable Hamiltonian system without viscoelastic terms. Then, by applying stochastic averaging, the system is further reduced to a partially averaged system of less dimension. The adjoint equation and maximum condition for the optimal control problem of the partially averaged system are derived by using the stochastic maximum principle, and the optimal bounded control force is determined from the maximum condition. Finally, the probability and statistics of the stationary response of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation (FPK) associated with the fully averaged Itô equation of the controlled system. An example is worked out to illustrate the proposed procedure and its effectiveness.